1. **State the problem:** Given that $a - b = b - c = 2$, find the value of $$\frac{(a - b)^2 + (b - c)^2}{(a - c)^2}.$$\n\n2. **Identify known values:** From the problem, we have $$a - b = 2$$ and $$b - c = 2.$$\n\n3. **Find $a - c$:** Using the property of subtraction, $$a - c = (a - b) + (b - c) = 2 + 2 = 4.$$\n\n4. **Substitute values into the expression:**\n$$\frac{(a - b)^2 + (b - c)^2}{(a - c)^2} = \frac{2^2 + 2^2}{4^2}.$$\n\n5. **Calculate numerator and denominator:**\n$$\frac{4 + 4}{16} = \frac{8}{16}.$$\n\n6. **Simplify the fraction:**\n$$\frac{\cancel{8}}{\cancel{16}} = \frac{1}{2}.$$\n\n**Final answer:** $$\frac{(a - b)^2 + (b - c)^2}{(a - c)^2} = \frac{1}{2}.$$